The game of keno uses 80 balls numbered 1 thru 80. Every game, the house draws 20 balls at random and displays their numbers on screens (called keno boards) located throughout the casino. The object of the game is for you the player to guess some of the numbers the house will draw.

You make your guesses by marking a keno ticket, a piece of paper with the numbers 1 thru 80 printed on it. Keno tickets are located at tables throughout the casino but are most readily found in the casino’s keno lounge: a room or area with chairs to sit and write. Usually crayons are provided for marking the ticket.

Before you go marking a ticket you need to decide how much money you want to gamble on the ticket. Near any stash of tickets you are likely to find several copies of the house’s keno brochure, which tells you what the standard bets and payouts are this year. At any point in time all casinos tend to have similar betting/payoff scales, but sometimes there are various “house specials” available. For the sake of simplicity, let’s assume for right now you avoid “specials”.

The top line, “Play 6 numbers” means this section of the brochure pertains to the payoffs you would be paid if you marked 6 numbers. “Numbers” are also referred to as “spots”. If you play 6 numbers you are said to be “playing a 6-spot”. You can play more or fewer than 6 spots but for now let’s stick with 6.

The term catch refers to how many of your (6 in this case) chosen numbers match what the house draws. The term “play $1” means you can bet $1 on your ticket. As the table shows, you can bet more than $1 if you’re in a real hurry to lose your money. There is no per cent advantage to betting more than the minimum; payoffs are simply scaled by the amount you bet. (In certain -rare- cases your return actually decreases when you increase your bet because the house has an upper limit of how much money per game they will pay off, independent of amount wagered.)

As the payoff table shows, if you play 6 numbers and catch all 6, a $1 ticket will return $1500. What are the odds of this event? A bit over 7500 to 1, as we will see in a later section. It does happen, but not often. (Mrs. Mello, across the street from me when I was in high school, hit a 6-spot in Reno one weekend.)

If you “only” catch 5 numbers the payoff is substantially less, and so on, down to a $1 payoff for catching any 3 of the 6 you selected. If you catch fewer than 3, your ticket is worthless. Of course this is just an example of a payoff schedule; that’s why you need to check the keno brochure to see what payoff scale the casino is using. A later section shows how to calculate the odds.

Note that this payoff is on a “for” basis rather than a “to” basis. Meaning if you collect, say, $8 for matching 4 numbers above, you’ve already paid $1 for the ticket so you’re only “winning” $7. Nevertheless it is standard keno terminology to say you “won $8”.

You can mark a ticket with anywhere from 1 to 15 (more in some places) numbers. The more numbers you mark the more you have to catch to win. The payoffs are set such that in dollar terms the house percentage is pretty much the same regardless of how many numbers you mark. You will find some variety in the minimum payoff so you can to some extent choose if you want many small wins or fewer, larger wins.

Sometimes when you mark a lot of numbers the casino pays off if you catch 0. This would be shown in the keno brochure.

Q:K2 How do I play a simple keno ticket?

A:K2 [John Hallyburton]

A basic example:

Let’s say you decide to play a $1 6-spot. Pick a blank keno ticket, grab a crayon and cross out your 6 choices with a plain X. On the right of the ticket write “$1” and beneath that the number “6” to indicate you are playing a 6-spot. Since you’ve already crossed out 6 numbers it’s kind of redundant to write “6” but this is used for cross-checking by the dealer, as well as being important when playing fancy combination tickets.

Having marked your ticket you now bring it to a dealer (also called a writer) at the front of the keno lounge. It’s a bit like standing in line at a bank since the dealer positions look much like teller windows. Hand your ticket and money to the dealer. He or she will make a copy of your ticket and give you the copy, retaining the original. Dealers don’t use crayons to make copies: some keno lounges use brushes and india ink while others use computers to generate copies. You are supposed to verify your copy before leaving the window because in case of a big win the house will verify that you actually marked those numbers on the original ticket you gave the dealer. They index and save the original tickets and will hunt through them if you catch a big winner. This is an anti-counterfeiting measure.

Once you have your copy you find someplace comfortable and wait for the rest of the players to be served. Eventually this happens and the house declares the game “closed”. All the original tickets are collected and bundled together someplace visible to a video camera, and the balls are mixed in the hopper. One of the dealers opens the portals and the chosen balls work their way out. One by one the dealer calls out the numbers and throws the switch that causes that number to light on the keno board. After the 20th and last number is chosen the dealers return to their stations. The few lucky winners rush to cash in while the rest of the players decide what numbers to pick for the next game. U.S. tax law requires winning tickets to be cashed immediately after the game; otherwise casinos could deduct unclaimed payouts as potential future liabilities. At least, this is what the casinos claim. Whether there actually is such a tax law on the books is an open question. Certainly casinos don’t want to keep original tickets and other records any longer than absolutely necessary, so a short time limit is obviously convenient for them.

If you have a losing ticket for a game and you want to play the same ticket for the next game, you don’t need to mark up another blank ticket. Just hand the losing ticket to the dealer along with the money and they will make you another copy. Actually this works for winning tickets, too. If you cash in a ticket that pays off, the dealer will probably ask “Want to play it again?” If you answer yes you’ll get back a new ticket and your winnings, less the cost of the ticket for the next game.

Occasionally I get email requests for “the keno formula” from people who want something to plug into a spreadsheet. There is no simple formula, though if you slog through this section you can come up with a complicated formula.

The proper buzzword for keno odds is “hypergeometric distribution”. But as usual, understanding the math is far less important than understanding how to apply it properly. First, let’s do the basics: if you mark N spots, the probability of hitting exactly K of them is given by the formula:

The expression C(X,Y) represents the number of possible ways to select Y items from a larger collection of X items, where order of selection is unimportant. Many calculators, spreadsheets and math libraries have a built-in facility for calculating this function. Both Lotus 1-2-3 ™ and Excel ™ name this funcion COMBIN(n,r); it is also known as the “binomial coefficient” function. (Caution: even if defined by your spreadsheet you may find the numbers involved too large to be handled by your spreadsheet program). Direct evaluation comes from the following formula:

X!
C(X,Y) = -----------------
Y! x (X-Y)!

… where “X!”, pronounced “X factorial”, is the product of all whole numbers from 1 to X. Thus 4! = 1 x 2 x 3 x 4 = 24. As a degenerate case, 0! = 1. So C(5,3) = 5!/(3!x2!) = 120/12 = 10. There are 10 ways to select 3 items from a bag of 5 items. Again, order of selection is unimportant. Note that N! = N x (N-1)!, for N>0. This can be useful in simplifying calculation.

Finally, 2 into 4 2 times. You can always divide all numbers in the bottom into the numbers on top. We are left with:

= 2 x 79 x 13 x 77 x 76 x 25
= 300500200

the number of ways to select 6 items from 80.

Notice if you select 6 items from a group of 80, you are “leaving” 74 items unselected. They form a group of their own! That means C(80,74) = C(80,6). Whenever you select a group, you actually select two groups, the “ins” and “outs”. In other words: C(X,Y) = C(X,X-Y). This amounts to switching the order of the multiplication of the bottom half of the fraction in the definition of C(X,Y).

For reference, C(80,20) = 3.535316142 x 10^18 or about 3 1/2 quintillion ways for the house to draw 20 balls. That number is so huge it is unlikely any random keno draw has ever happened twice in all of history. Well, maybe one repetition somewhere, if you’re generous enough estimating how many games have taken place over time.

Let’s use the formula to calculate our chances for hitting a 6-spot:

p(6,6) = C(6,6) * C(74,14) / C(80,20) ~= 0.00013 or 1 in 7753.

Rarely do we hit all 6. Let’s calculate the whole 6-spot table from the above formula. Note these numbers are independent of the house payoff as they are merely the probability of an event happening, regardless of whether any money is wagered.

The total should always be 1, or very close due to rounding, since one of the above outcomes will happen.

To find the probability of one of several outcomes, you add the numbers for each entry. In the above 6-spot table the chances of catching “exactly 0 OR exactly 1” are 0.52951. Meaning more than half the time you’ll catch at most one number on your 6-spot ticket. Similarly, roughly 1 in 6 tickets will be a winner according to the payoff table presented above. (Since Catch3+Catch4+Catch5+Catch6 probabilities total 0.16159 or 1/6.19).

Of course the chance of having a winning ticket is not as interesting as knowing what the expected return is. This is calculated by adding together the expected return for every possible outcome. Now for any possible outcome you can calculate the expected return for that outcome by multiplying the payoff for that outcome by the probability of that outcome. This leads directly to the following set of calculations, using the 6-spot payoff and odds already presented:

For this payoff schedule you can expect to receive a return of 71 cents for every dollar bet. The house advantage is a whopping 29%. While this is a huge advantage for the house, remember that the overhead of keno is also the highest of any casino game. There is plenty of floorspace devoted to the game, a large number of dealers, and relatively small bets are the norm. It would not be profitable for a casino to run keno at a much lower “take”. At least, this is what the casinos say when asked about the high take. While it seems reasonable, nobody in the business has actually offered any proof that this is the case.

Q:K4 What are some strategies for marking tickets?

A:K4 [John Hallyburton]

There are a variety of strategies for playing keno. None of them provide you with any advantage, but they can be fun to play. Some of them can also be mighty expensive.

As with roulette you can “chase the old man”, meaning play those numbers that seem to be coming up more often than the others. The theory is that they will keep coming up again in the future. The reality is they don’t come up any more than chance dictates. Or you can “let the old man chase you”, meaning play numbers that haven’t come up in recent games, on the theory that they’ll start coming up in order to make the long-run results for each number even out. Again, the reality is that the balls have no memory. Number 47 may not have come up in the last 10 games, but that has no predictive value. In such a case the molecules that make up ball number 47 do not strain themselves to “even the score”.

The keno playing card is divided into an upper and lower half. My uncle Dave would always be sure to spread his picks evenly across both halves in order to “play the whole card”. His theory was that if you just picked 6 numbers from say the lower half then you wouldn’t catch as many as if you picked 3 numbers from the lower half and 3 from the upper half. After all, most every game will see about half the house numbers drawn in the upper part and half drawn in the lower part. But the reality is that it doesn’t matter, the balls don’t care what the playing cards look like. The house could print every card differently. A set of numbers that produces a balanced card for one configuration might be highly unbalanced in another. This will not cause one set of numbers to be drawn more often than any other set.

While not exactly a strategy, some players may find it entertaining to play somebody else’s losing ticket. If the guy next to you gets up to leave, take his losing ticket to the dealer and say “Play it again”. This takes a certain amount of timing, judgement and chutzpah (a by-product of several Rumple Minze’s on the house), but can be fun if pulled off well.

Q:K5 What’s a keno runner?

A:K5 [John Hallyburton]

Most casinos offer “keno runners” as a courtesy to players who happen to be in the dining room or poker table, but still want to play keno. “Keno girls” are often dressed much like cocktail waitresses and carry trays with blank tickets, crayons and spare change so that patrons can play keno anywhere in the casino.

A typical comped lunch in Las Vegas takes just about long enough to play (and lose) four to five keno games.

Usually keno runners cruise the casino calling out “keno?” to no one in particular. If you wish to engage the services of a runner, merely answer by saying the word “keno!” in a louder tone. The runner will stop by your table, wait for you to mark and pay for a keno ticket, give you any change, and continue cruising.

The runner will take your ticket to the keno lounge and have a dealer make a copy just as you would. Usually the keno runners are the last to be served before the game is closed. The runners wait for the numbers to be drawn and then return to their customers with their tickets and winnings. They will also have a keno ticket with holes punched for every number drawn, so you can lay the punched ticket over your ticket and count spots.

While the casino makes every effort to ensure all runners make it back to the lounge before the game is closed, they cannot guarantee that your ticket will get played in the next game, nor that the dealer will copy your ticket correctly. Surely there are apocryphal stories of runners who were late to the lounge and caused a player to miss a 10-spot. You have to be willing to accept that additional risk if you use a keno runner.

Fellow r.g.o-g’er Dave Everett adds:

“Just about everyone who performs a service for people in a casino works for tips, and the keno runners are no exceptions. If a keno runner services your bets for you, it is customary to tip a small amount even if you don’t win. You are not expected to tip every game, but, say you are having lunch, and the keno runner hits your table and services your bets 5 times, and you never win. You should tip a dollar. You know when you’re just about done eating. When the runner comes to your table with and checks your losing ticket and asks if you want to play it again, that’s the appropriate time to say “No, thanks; here’s a dollar for your trouble.” Personally, I tip a dollar the very first time the runner comes back (and not again if I don’t win anything). I’ve found that I get excellent runner service that way.

“Now if you hit anything substantial, the keno runner will expect something. You don’t have to be super-generous. My personal rule of thumb is $2 for a $10 to $40 win; 5% for larger wins up to $200. You can get away with a little less, but don’t stiff them. Yes, tipping decreases your expectation, but if you employ a keno runner, you are buying a service. If you don’t want to tip, run your own tickets up to the window. If you hit big enough to tip the runner, tip her/him when (s)he brings you your winnings. I *guarantee* you’ll get excellent service thenceforth. They do share information, too. If you tip one runner, (s)he will pass on the information to the relief or replacement runner. If you don’t, that info will get passed on, too.”

Q:K6 Can I play anything besides individual N-spot tickets?

A:K6 [John Hallyburton]

Oh, yes. There are combinations and way tickets.

A “combination” is a ticket that combines several different choices. For example you can mark, ohh, 3 numbers on the top half of a ticket and, say, 4 numbers on the bottom half of the same ticket. Suppose you want to play the top 3 numbers as a 3-spot, the bottom 4 numbers as a 4-spot and the whole ticket as a 7-spot. Now you can, if you want, mark 3 separate tickets and pay $1 each, and cash any winners individually. OR you can play a combination ticket, which is just a special way to combine those three tickets into one. Here’s how: mark your 7 numbers and draw a thick line separating the 3 top numbers and the bottom 4. Then over on the right of the ticket you write: “3$”; below that “1/3” (shorthand for 1 3-spot); below that (“1/4”, for 1 4-spot) and below that “1/7”, for (you guessed it) 1 7-spot. Then below THAT you write “$1”, meaning you are betting $1 on each of those combinations. This is actually 3 separate wagers and if you win you will be paid off as if you had submitted 3 separate tickets. It’s just keno shorthand.

Speaking of shorthand, the notation: “$3″;”1/3″;”1/4″;”1/7″;”$1” is this FAQ entry’s notation for writing those numbers in a column off to the right of the card, as in

$3
1/3
1/4
1/7
$1

You can get even more exotic. Say in addition to the 3- and 4-spots you want to play a 2-spot in the lower left. Now a simple horizontal line isn’t enough to separate the groups of numbers you choose. What you would do is circle the groups of 2, 3 and 4 to clarify how you want to group your ticket. Then off to the right you explain how you are betting. Are you playing a 2-spot, 3-spot and 4-spot? How about the 5-spot, 6-spot and 7-spot that comes from combining the 2-, 3-, and 4-spots in various groupings? How about the 9-spot that comes from playing all 9 numbers at once? Once again, you indicate what you are playing by making notations on the right of the ticket. You don’t have to play everything. You can play the 2-spot, 3-spot, 4-spot and 7-spot by marking your ticket with: “$4”; “1/2”; “1/3”; “1/4”; “1/7”; “$1”. But beware: if your 2-spot and 4-spot come up, you’ll get paid the $85 or so total for catching the 2-and 4-spot but you won’t get paid for catching the 6-spot formed by combining them because you dind’t play that option. You would have had to pay another $1 and designate “1/6” on your ticket.

If you mark a single spot and circle it, it is sometimes called a “king number” and is usually combined with other groups or even other king numbers. But it’s basically a group-of-one.

Even more complicated are “way tickets” which are essentially combination tickets that involve a large number of uniform choices intertwined in all possible ways. A simple example involves picking, say, 5 sets of two numbers each. Maybe you choose 11-22 (mark and circle these two), 4-25 (mark and circle these two), 38-40 (mark and circle these), 64-65 (mark and circle these) and 76-77, also marked and circled.

Now what you want to play is every possible 6-spot that can be formed by combining the circled numbers. With 5 groups of two numbers, noting you need 3 groups of 2 to form 6 numbers, there are C(5,3) = 10 possible ways to form 6-spots out of those groups of two. So far you’ve got a ticket with 5 circles of 2 numbers each. To the right of the ticket you write: “$10″;”10/6″;”$1”. That is, you are paying $10 for the ticket, playing 10 ways of 6 spots at $1 per way. For another dollar you could have also played the 10-spot that is the total collection of 10 numbers that you circled. That ticket would be marked “$11″;”10/6”; “1/X”;”$1″. The “X” is keno notation for “10” when it denotes the number of spots being played. (No, “V” is not used for “5”).

Beyond complicated, into the realm of hairy, is the 190-way 8-spot ticket. Nearly every keno brochure features this to entice players into what looks like it must be a sure thing. The player draws a horizontal line to divide the card into upper and lower halves. Then draw a vertical line between each column as well. This has the effect of dividing the card into 20 columns of 4 numbers each, with the intent of playing all possible ways of forming 8-spot tickets from pairs of columns of 4 numbers each. Since it takes two columns to form an 8-spot, and we have 20 columns, there are a total of C(20,2) = 190 ways to combine 2 columns, i.e., create 8-spots.

If you were to play this ticket at the $1 rate it would cost you $190 per game. You are welcome to make that wager, but the casinos usually allow you to bet less than the nominal minimum when you are playing way tickets. For example, many casinos will let you bet 25 cents per way on this type of ticket. At the 25-cent level you would write this up as:

(ticket with $47.50
lines drawn 190/8
all over it) 25c

Since you are playing 25 cents per way, any payoffs would be at one-fourth of the $1 payoff scale. When the numbers are drawn for this ticket you hope an entire column of 4 lights up, then it’s just a matter of waiting to see how much you’ll collect.

As you can see, way tickets can be both expensive and exciting. But like combination tickets they are really nothing more than keno shorthand for a large number of individual tickets. Consequently they offer no financial advantage or disadvantage over regular tickets.

Q:K7 What’s this “red game” and “green game” I see some places?

A:K7 [John Hallyburton]

Some casinos will run multiple games to entice players to wager more. You mark your ticket and tell the dealer to enter it in the “red game”, “green game” or both. Basically it’s a way to increase action (wagering) without having to increase floorspace and staff correspondingly. There are two separate sets of balls and two keno boards, but only one keno lounge and set of dealers. The casino figures you won’t want to just play one of the games because you “know” if you play just the “red game” your numbers will come up only on the “green game”. So players tend to bet on both games, increasing the house’s take. Mathematically there’s nothing special about playing both games, though sometimes casinos offer prizes for hitting on both games. You can calculate the odds of hitting on both games by multiplying combinations that pay off. For example, the chance of catching 4 out of 6 is 0.02854, so the chance of catching 4 out of 6 in 2 simultaneous games is 0.02854 x 0.02854 ~= 0.0008 or 1 in 1228.

Q:K8 What about video keno?

A:K8 [John Hallyburton]

Video keno offers the same odds as regular keno, but the payoff scale may vary from the casino norm. You can use the math section to calculate your return from video keno and compare it with the return from the casino’s live game. Often you can find video keno games where a casino doesn’t have a live game, and the bet size is often lower. Nickel VK games are not uncommon.

Q:K9 Got any good keno stories?

A:K9 [John Hallyburton]

Here’s one from the “Believe it or not” school, and a true one.

Back in the ’60s keno games featured a maximum payout of $25,000 per game. One Friday night at Binion’s Horseshoe they had back-to-back $25,000 winners. Word got around Fremont street mighty fast. By 6am Saturday morning Binion’s had made far more than $50,000 profit from a keno game that had suddenly become the most popular pastime in downtown Las Vegas.

More recently a Canadian casino was “hit” for a fair take by someone who noticed the machines always produced the same numbers in the morning. (Does anyone want to send me the details?) Basically the machine was missing a computer chip and the casino had turned the machine off at night. So in the morning the random numbers were always started off from the same base instead of keying off something truly random like the time it took for a player to feed a coin. But the guy who discovered this got a bit too greedy and hit too many jackpots, and the truth came out.